Benford＇s Law and β-Decay Half-Lives

The experimental values of 2059 β-decay half-lives are systematically analyzed and investigated. We have found that they are in satisfactory agreement with Benford＇s law, which states that the frequency of occurrence of each figure, 1-9, as the first significant digit in a surprisingly large number of different data sets follows a logarithmic distribution favoring the smaller ones. Benford＇s logarithmic distribution of β-deeay half-lives can be explained in terms of Neweomb＇s justification of Benford＇s law and empirical exponential law of β-decay half-lives. Moreover, we test the calculated values of 6721 β-decay half-lives with the aid of Benford＇s law. This indicates that Benford＇s law is useful for theoretical physicists to test their methods for calculating β-decay half-lives.     

Products of random variables and the first digit phenomenon

We provide conditions on dependent and on non-stationary random variables $X_n$ ensuring that the mantissa of the sequence of products $\left({\prod }_1^n{X}_k\right)$ is almost surely distributed following Benford’s law or converges in distribution to Benford’s law. This is achieved through proving new generalizations of Lévy’s and Robbins’s results on distribution modulo 1 of sums of independent random variables.     

First Digits’ Shannon Entropy

Related to the letters of an alphabet, entropy means the average number of binary digits required for the transmission of one character. Checking tables of statistical data, one finds that, in the first position of the numbers, the digits 1 to 9 occur with different frequencies. Correspondingly, from these probabilities, a value for the Shannon entropy H can be determined as well. Although in many cases, the Newcomb–Benford Law applies, distributions have been found where the 1 in the first position occurs up to more than 40 times as frequently as the 9. In this case, the probability of the occurrence of a particular first digit can be derived from a power function with a negative exponent p > 1. While the entropy of the first digits following an NB distribution amounts to H = 2.88, for other data distributions (diameters of craters on Venus or the weight of fragments of crushed minerals), entropy values of 2.76 and 2.04 bits per digit have been found.